Characteristic of Stress Distribution at a Vertex in Orthotropic Piezo-ceramic Bi-material Bonded Joints

Transcription

1 ceedngs f the Intenatnal Cnfeence n Mechancal ngneeng Renewable negy 7 (ICMR7) Decembe 7 Chttagng Bangladesh ICMR7-I-57 Chaactestc f Stess Dstbtn at a Vetex n Othtpc ez-ceamc B-mateal Bnded nts Md. ShahdlIslam A. T. M. Mashd fess Dept. f Mechancal ngneeng Khlna Unvesty f ngneeng & Technlgy Bangladesh Dept. f Mechancal ngneeng Khlna Unvesty f ngneeng & Technlgy Khlna-93 Bangladesh shahdlbtk@gmal.cm * masd.an9@gmal.cm Abstact-Mateal scence smlatn s gettng ppla f ts cst beneft at. Wth the develpment f new mateals eplacng ld cstly mateal wth the new ne s gng t help the next geneatn technlgy f smat mateal. Smlatn pvdes a bette way f the test as t nly eqes engh pcessng pwe f the cmpte t fnsh the calclatn. It s less expensve than the eal test bt can gve almst accate eslts t pedct the fte f the tests. ez-ceamc mateal has ganed me attentn n ecent yeas f the ppetes hge applcatn n senss tansdces. S t s f geat mptance t knw hw pezelectc mateal behaves nde mechancal stess especally when they ae bnded wth anthe ne. Many eseaches have been nvestgated the chaactestc f stess feld n pez-ceamc bned jnts. Bt the stess cncentatn nea the vetex f nteface n bnded jnt s nt clea ntl nw. In ths pape chaactestc f stess dstbtn at a vetex f nteface s nvestgated sng egen analyss based n Fnte lement Methd (FM). Fm the smlatn eslt t s bseved that the hgh stess cncentatn ccs nea the vetex f nteface n thtpc pez-ceamc b-mateal Bnded nts. Keywds: ez-ceamcs Othtpc FM gen Analyss B-mateal. INTRODUCTION In the mateal scence pezelectc mateals have dawn attentn n the ecent yeas de t ts ppetes. ezelectcty s the electc chage that accmlates n cetan sld mateals sch as cystals cetan ceamcs blgcal matte sch as bne DNA vas ptens n espnse t appled mechancal stess. F ts ppety t s wdely sed f tansdce sens []. A pezelectcsens sadevcethatsesthepeze lectceffect t mease changes n pesse acceleatn tempeate stan fce by cnvetng them t an electcal chage. When pesse fce changes electcty geneate by calbatng them t a knwn vale t ae sefl t se them as sens.a tansdce s any devce sed t cnvet enegy fm ne fm t anthe; typcally when cnvetng npt enegy nt tpt enegy. F tansdctn t cc a change fm ne fm f enegy mst als take place sch as a cnvesn fm mechancal t electcal enegy vce vesa[]. Dng defmatnez-electc mateal pdces sgnfcant amnt f electcty t affect ts dffeent behav. S dffeent pez-electc d-electc ppetes ae nclded n the analyss. Adhesve bndng has been nceasngly sed n jnng epang lad-cayng stctal cmpnents becase f ts chaactestcs. Geneally adhesve bnd s lghte than any the me nfm effcent lad tansfe nt the patch can edce the sk f hgh stess cncentatns. Ths leads t the wde se f bnded epas nstead f sng veted epas n acaft stctes. One f the cmmn examples f bnded jnt s the sngle step btt jnt T-jnt. xstng sngle step jnts ae made f tw sbstates jned by sng the mechancal cnnectn methd chemcal cnnectn methd sld phase bndng pcess [3]. When tw pez-ceamc mateals ae jned tgethe placed nde a tensle fce a hgh stess cncentatn ccs nea the vetex f the jnts. Many eseaches have been nvestgated the effect f stess feld n pez-ceamc bned jnts. Bt the chaactestcs f stess cncentatn nea the vetex f nteface ae nt clea ntl nw. In ths pape chaactestc f stess dstbtn at a vetex f nteface n bnded jnts s nvestgated sng egen analyss based n FM. The egen eqatn s sed f nvestgatng the de f stess snglaty the angla fnctn f elastc dsplacement electc ptental stess electc dsplacement.. FORMULA OF ANALYSIS In the absence f bdy fces fee chages the eqlbm eqatns f pezelectc mateals ae expessed as fllws[]: ICMR7

2 j j d. () The cnstttve elatns ae shwn as fllws: j cjkl kl ekjk d ekl kl k k. () The elastc stan-dsplacement electc feld-ptental elatns ae pesented as fllws: j j j (3) whee j k l = 3 j d j ae the cmpnent f stess electc dsplacementstan elastc dsplacement electc feld electc ptental espectvely. F thtpc mateal the cnstctve elatn s expessed n the fllwng fm. c e d e () In Catesan cdnate the cnstttve eqatn edced t the matx fm as fllws. c c c3 c c c3 33 c 3 c3 c c 3 3 c55 3 c66 e3 e3 e33 (5) e 3 5 e d e5 33 d e d e e e 3 (6) 33 3 The cmbned fm f q. () s expessed as fllws. d c e T D D e (7) Sngla pnt x Fg.: lement gemety natal c-dnates at a fee edge sngla pnt Fge epesents the gemety f a typcal case whee a sngla stess state ccs at the pnt. The egn sndng the sngla pnt s dvded nt a nmbe f qadatc pyamdal elements wth a smmt wth each element beng lcated n sphecal c-dnates by ts ndes t. A pnt n the element can be lcated sng the sngla tansfmatn by the fllwng elatns []. H H () (9) Wheepegen vale = / the dstance fm the sngla pnt H ndcates the shape fnctn whch s wtten as; H H H3 H H 5 H6 H 7 H ae the ndal vales f the angla c-dnates ae natal c-dnates f the element whse anges ae shwn n Fg.. The elastc dsplacement electc ptental feld n the element s expessed as fllws: whee H H () z s the elastc dsplacement at espectvely s the electc ptental at espectvely. In de t smplfy the ntatn the fllwng eqatn can be defned. y ICMR7

3 ICMR7 ) ( (5) Usng the q. ()q. () can be expessed as fllws: () whee H k H k k The acban matx elatng the sphecal cdnates t the natal cdnates s gven belw: ) (3 H H H H () q. (3) shws that thee s n dependence between the adal cdnate the angla cdnate. Fm q. (3) a sb-matx s extacted as fllws: ) ( H H H H () The stan electc ptental eqatn s btaned fm qs. () (9) qs. () () by sng the chan le f dffeentatn. The stan n a sphecal cdnate system []: H p H H H ct sn ct H H sn H sn H sn ct ct H H H sn H sn H sn p p H sn H sn H p p H ) (5 H H The electc ptental n a sphecal cdnate system: H p H H (6) sn sn sn H H The spescpt - n the matx [ ] epesents the nvese matx. qs. (5)(6) nw can be smmazed as fllws: ) (7 B B

4 Whee * T * T B p pb a B b q.(7) epesents the stans theefe the stesses ae pptnal t p-. The case whee <p< defnes a sngla stess state at the vetex f the element. The element depcted n Fg. mst satsfy the pncple f vtal wk n de t be n eqlbm that s j b j jjd T d f d () (6) q (3) Whee T * epesents the tactn at the te bnday. Ths eqatn can be tansfmed nt a matx fm wth the help f qs. () (9) as fllws: T ε sn ddd T T H sn dd (9) - - d Whee epesent the detemnant f the matces [] [ ] espectvely T s epesented by the fllwng eqatn. T d d d () The elatn between stess electc dsplacement wth stan electc feld s as fllws: D () whee D epesents the mateal cnstants matx. The 99 matx [D] s expessed n the sphecal cdnate system. The matx [D] s evalated at each Gass pnt dng the nmecal evalatn f the ntegals. The mateal ppetes n the ectangla cdnate system (x y z) ae tansfmed t the sphecal cdnate system ( ). Whee epesent the sphecal cdnates f any Gass pnt. The egen eqatn was fmlated f detemnng the de f stess snglaty as fllws [5]: whee U Mateal p A pb C U () y (6) O A k k k k s a sa B s C k k s In q. ()p epesents the chaactestc t whch s elated t the de f snglaty as= -p. [A] [B] [C] ae matces cmpsed f mateal ppetes {U} epesents the elastc dsplacement electcptental vect. The elastc dsplacement electc ptental stess electc dsplacement felds n stess snglaty egn ae expessed by the fllwng eqatns. By dffeentatng the abve tw eqatns get the angla fnctn f stan electc feld eqatn espectvely. The stess electc dsplacement dstbtn eqatns n the stess snglaty egn can be expessed as fllws. j d fj Kj l c F () sc b Whee epesents the dstance fm the stess sngla pnt b j ( ) the angla fnctn f elastc dsplacementq( ) the angla fnctn f electc ptental f j ( ) the angla fnctn f stess dstbtnl ( )the angla fnctn f electc dsplacement K j the ntensty f snglaty F the ntensty f electc feld λ the de f stess snglaty. Angla fnctns f stess electc dsplacement cmpnents btaned fm gen analyss n () ae examned. 3. MODL AND MATRIAL OF ANALYSIS The mdel f 3D thtpc pezelectc bnded jnts f egenanalyss s shwn n Fg. (a). The angle f ae 9 espectvely.in egenanalyss a mesh dvsn f the jnt s needed f the analyss. The mesh develped n - plane s shwn n Fg. (b) whee the sface f a nt sphee s sbdvded nt = f gh mesh = f fne mesh. z / StessSnglaty Lne Mateal Vetex (a) (b) Fg.:(a) 3D pezelectc jnts wth cne n x y z c-dnate (b)a mesh n the develped - plane x / Mateal Mateal sb Inteface Snglaty Lne ICMR7

5 The mateal ppetes f thtpc pezelectc mateals ae shwn n the Table. VDN(lyvnyldene Flde) BSN(Bam Sdm Nbate)ae sed f ppe lwe matealsn the analyss. Table : Mateal ppetes f VDF BSN lastc cnstant N/m Mateal C C C 3 C C 3 C 33 C C 55 C 66 VDF BSN Mateal ezelectc cnstant (C/m ) Delectc cnstant ( - C/Vm) e 5 e e 3 e 3 e 33 χ χ χ 33 VDF BSN NUMRICAL RSULT AND DISCUSSION Slvng egen eqatn yelds many ts p egen vects cespndng t each egen vale ae btaned. Hweve f the t s wthn the ange f <p< ths fact ndcates that the stess feld has snglaty. The vale f de f snglaty at the snglaty cne s.577 In the pesent pape the de f snglaty s nvestgated vayng the mateal cnstants. A at f mateal cnstants t a specfed mateal cnstant s ntdced as fllws. Fg. 5: Vaatn f de f snglaty aganst χ/χ n In these thee gaphs t s seen that by changng elastc ppetes the de f snglaty s changng sgnfcantly. Bt when changng the vale f pezelectc d-electc ppetes f thtpc mateal shws the small effect n the de f snglaty. S t can be sad that changng f elastc ppetes have the sgnfcant effect n snglaty. Angla fnctns btaned fm egen eqatn q. () ae examned. Dstbtns fangla fnctn f elastc dsplacement electc ptental n a - plane ae shwn n Fg. 69 angla fnctn f stess electc dsplacement ae n shwn n Fgs. 5. b Whee epesent the efeental pezelectc delectc elastc cnstants espectvely f VDF BSN.e χ c epesent the new pezelectc delectc elastc cnstants espectvely. The vale f S vaes fm. t. T plt the vale lgathmc vale f the S s taken. Fg 6: Dstbtn f angla fnctn f dsplacement cmpnents b. b θ Fg. 3: Vaatn f de f snglaty aganst c/c n Fg 7: Dstbtn f angla fnctn f dsplacement cmpnents b θ. b ϕ Fg. : Vaatn f de f snglaty aganst e/e n Fg. : Dstbtn f angla fnctn f dsplacement cmpnents b ϕ. ICMR7

6 q Fges 3 5 shw the 3D dstbtn f angla fnctn f electc dsplacement n-plane f =.577. All these gaphs shw the angla fnctn f electc dsplacement s cntns at the nteface f the dssmla mateal jnt. Fg. 9:Dstbtn fthe angla fnctn f electc ptental q. All f these gaph shw that the angla fnctnsae cntns at the nteface. The nteface f the jnt s at = 9. Fges shw the 3D dstbtn f angla fnctn f stess n -plane f =.577. All these gaphs shw the angla fnctn f stess have the hghe vale at the nteface edge f the dssmla mateal jnt. Fg. 3: Dstbtn f angla fnctn f electc dsplacement cmpnents l. f θθ Fg. : Dstbtn f angla fnctn f electc dsplacement cmpnents l θ. Fg. : Dstbtn f angla fnctn f stess cmpnents f θθ. f θ Fg. : Dstbtn f angla fnctn f stess cmpnents f θ. Fg 5: Dstbtn f angla fnctn f electc dsplacement cmpnents l ϕ. f θ Fg. : Dstbtn f angla fnctn f stess cmpnents f θϕ Fg. 6: Dstbtn f nmalzed angla fnctn f stess cmpnents f j alng ϕ when θ=9 The nmalzed angla fnctn f stess s shwn ICMR7

7 n Fg.6 f =.577. The angla fnctn f stess aganst the angleat= 9s pltted. The stess snglaty lnes ae at the fee edge f the mateal jnt. The gaph shws that the vale f stess nceases apdly nea the fee edge than the nne ptn f the jnts. The vale f f f s ne f s ze nea = 5. Afte that the vales f angla fnctn f stess ae nceased. Nea fee edge f the jnt has the hghest vale f angla fnctn. S thee s a pssblty f debndng delamnatnccs nea the nteface edge f the jnt. [] Md. Shahdl Islam Kgch Hde Chaactestcs f sngla stess dstbtn at a vetex n tansvesely stpc pezelectc dssmla mateal jnts nal f Sld Mechancs Mateals ngneeng vl. sse 7 pp. -6. [5] H. Kgch Stess snglaty analyss n thee-dmensnal bndedstcte Tans. SM A 7 n. 7 pp [6]. Q.XU Y. Mth Snglaty at the nteface edge f bnded tansvesely stpc pezelectc dssmla mateal SM Intenatnal nal Sees A Vl. N.. [7] T. Ikeda H. Ha M. Abe M. Chba N. Myazak Stess ntenstyfact analyss f an ntefacal cne between pezelectc bmatealsn a tw dmensnal stcte sng H-ntegal Methd ceedngs fthe ASM InteACK pp NOMNCLATUR Fg. 7: Dstbtn f nmalzed angla fnctn f electcal dsplacements cmpnentsl alng ϕ when θ=9 The nmalzed angla fnctn f electc dsplacement s shwn n fg. 7 f =.577. The angla fnctn f electc dsplacement aganst the angleat= 9 s pltted. The shape f the cve s same as angla fnctn f stess cve. The gaph als shws that the vale f electc dsplacement nceases apdly nea the fee edge than the nne ptn f the jnts. The vale f l l s ne l s ze nea = 5. Afte that the vales f angla fnctn f electc dsplacement ae nceased. Symbl Meanng Unt j Stesstens Ma d lectc dsplacement C/m vect kl Stan tens m/m k lectc feld N/C c jkl lastc cnstant N/m e kj (e kl ) ezelectc cnstant C/m k lectc pemttvty C/Vm (delectc cnstant) lastc dsplacement m lectc ptental V 5. CONCLUSION An egen analyss nea the vetex f thtpc pezelectc dssmla jnt was pesented. Angla fnctns f stess electc dsplacement at snglaty cne wee deved fm egen analyss sng a fnte element methd.fm the smlatn eslts the fllwng cnclsns can be dawn f the Othtpcpezelectc mateal jnts. (a) lastc cnstant affects the de f snglaty me sgnfcantly. (b) Lage vale f the angla fnctn f stess electc dsplacement ccs at the nteface edge than the nne ptn f the jnt. (c) It ssggested that debndng cack f the nteface may ccnea the nteface edge f the jnts. 6. RFRNCS [] M. S. Islam K. Hde Investgatn f Ode f Snglaty n 3D Tansvesely Istpc ezelectc Bmateal nts by FM nal f Ccts Systems Cmptes Wld Scentfc blshng Cmpanyvl. n. pp. 5-. [] [3] ICMR7